Digital Communications I Modulation and Coding Course Spring

Digital Communications I: Modulation and Coding Course Spring - 2013 Jeffrey N. Denenberg Lecture 3 b: Detection and Signal Spaces

Last time we talked about: Receiver structure Impact of AWGN and ISI on the transmitted signal Optimum filter to maximize SNR Matched filter receiver and Correlator receiver Lecture 3 b 2

Receiver job Demodulation and sampling: Waveform recovery and preparing the received signal for detection: Improving the signal power to the noise power (SNR) using matched filter Reducing ISI using equalizer Sampling the recovered waveform Detection: Estimate the transmitted symbol based on the received sample Lecture 4 3

Receiver structure Digital Receiver Step 1 – waveform to sample transformation Step 2 – decision making Demodulate & Sample Frequency down-conversion Receiving filter Threshold comparison Equalizing filter Compensation for channel induced ISI For bandpass signals Received waveform Detect Baseband pulse (possibly distored) Lecture 4 Baseband pulse 4 Sample (test statistic)

Implementation of matched filter receiver Bank of M matched filters Matched filter output: Observation vector Lecture 4 5

Implementation of correlator receiver Bank of M correlators Correlators output: Observation vector Lecture 4 6

Today, we are going to talk about: Detection: Estimate the transmitted symbol based on the received sample Signal space used for detection Orthogonal N-dimensional space Signal to waveform transformation and vice versa Lecture 4 7

Signal space What is a signal space? Why do we need a signal space? Vector representations of signals in an N-dimensional orthogonal space It is a means to convert signals to vectors and vice versa. It is a means to calculate signals energy and Euclidean distances between signals. Why are we interested in Euclidean distances between signals? For detection purposes: The received signal is transformed to a received vector. The signal which has the minimum Euclidean distance to the received signal is estimated as the transmitted signal. Lecture 4 8

Schematic example of a signal space Transmitted signal alternatives Received signal at matched filter output Lecture 4 9

Signal space To form a signal space, first we need to know the inner product between two signals (functions): Inner (scalar) product: Analogous to the “dot” product of discrete n-space vectors = cross-correlation between x(t) and y(t) Properties of inner product: Lecture 4 10

Signal space … The distance in signal space is measure by calculating the norm. What is norm? Norm of a signal: = “length” or amplitude of x(t) Norm between two signals: We refer to the norm between two signals as the Euclidean distance between two signals. Lecture 4 11

Example of distances in signal space The Euclidean distance between signals z(t) and s(t): Lecture 4 12

Orthogonal signal space N-dimensional orthogonal signal space is characterized by N linearly independent functions called basis functions. The basis functions must satisfy the orthogonality condition where If all , the signal space is orthonormal. See my notes on Fourier Series Lecture 4 13

Example of an orthonormal basis Example: 2 -dimensional orthonormal signal space 0 Example: 1 -dimensional orthonormal signal space 0 0 T t Lecture 4 14

Signal space … Any arbitrary finite set of waveforms where each member of the set is of duration T, can be expressed as a linear combination of N orthonogal waveforms where Vector representation of waveform Lecture 4 Waveform energy 15

Signal space … Waveform to vector conversion Lecture 4 Vector to waveform conversion 16

Example of projecting signals to an orthonormal signal space Transmitted signal alternatives Lecture 4 17

Signal space – cont’d To find an orthonormal basis functions for a given set of signals, the Gram-Schmidt procedure can be used. Gram-Schmidt procedure: Given a signal set , compute an orthonormal basis 1. Define 2. For compute If let If , do not assign any basis function. 3. Renumber the basis functions such that basis is This is only necessary if Note that Lecture 4 for any i in step 2. 18

Example of Gram-Schmidt procedure Find the basis functions and plot the signal space for the following transmitted signals: 0 0 T T t t Using Gram-Schmidt procedure: 1 2 0 T t -A Lecture 4 0 19 A

Implementation of the matched filter receiver Bank of N matched filters Observation vector Lecture 4 20

Implementation of the correlator receiver Bank of N correlators Observation vector Lecture 4 21

Example of matched filter receivers using basic functions T 0 0 T t 0 t T t 1 matched filter 0 T t Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to the transmitted signal. Lecture 4 22

White noise in the orthonormal signal space AWGN, n(t), can be expressed as Noise projected on the signal space which impacts the detection process. Noise outside of the signal space Vector representation of independent zero-mean Gaussain random variables with variance Lecture 4 23